ON THE.MOTION OF PENDULUMS. 25 where ^, ^2 are the integrals of the equations -x-O............(22), W' / J2 da 1 _^__1 \dx* d'sr' ta dtff fjf dt. 11. By means of the last three equations, the expression for tip obtained from (16) and (17) is greatly simplified. We get, in the first place, \dx *<fa * v but by adding together equations (22) and (23), and taking account of (21), we get =_ ~~ rfcr2 «r cfe /// On substituting in (24), it will be found that all the terms in the right-hand member of the equation destroy one another, except those which contain dty/dt and d^Jdt, and the equation is reduced to dp __ p dx is dtdvr ' by JD, our equation becomes which gives by the separation of symbols so that d^/d* is composed of two parts, which are separately the integrals of (22), (23). Hence we have for the integral of (200 ^=^1 + ^2 + ^1 ^ being a function of x and tjj without t which satisfies the equation D2SI> = 0. For the equations (22), (23) will not be altered if we put f^dt, J^^dt for fa, fa, the arbitrary functions which would arise from the integration with respect to t being supposed to be included in SI/". The function ty, which taken by itself can only correspond to steady motion, is excluded from the problem under consideration by the condition of periodicity. But we may even, independently of this condition, regard (21) as the complete integral of (20'), provided we suppose included in (21) terms which would be obtained by supposing \jf at first to vary slowly with the time, employing the integrals of (22) and (23), and then making the rate of variation diminish indefinitely. By treating the symbolical expression in the right-hand member of equation (a) as a vanishing fraction, djdt being supposed to vanish, we obtain in fact D~2 0 ; so that under the convention just mentioned the function SI> may be supposed to be included in fa + fa. The same remarks will apply to the equation in Section III. which answers to (20').